A008594 -id:A008594 - cp's OEIS frontend

A304158 a(n) is the second Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference (Fig. 3).

24, 84, 144, 204, 264, 324, 384, 444, 504, 564, 624, 684, 744, 804, 864, 924, 984, 1044, 1104, 1164, 1224, 1284, 1344, 1404, 1464, 1524, 1584, 1644, 1704, 1764, 1824, 1884, 1944, 2004, 2064, 2124, 2184, 2244, 2304, 2364, 2424, 2484, 2544, 2604, 2664, 2724, 2784, 2844, 2904, 2964

Offset: 1

Emeric Deutsch, May 08 2018

The second Zagreb index of a simple connected graph is the sum of the degree products d(i)d(j) over all edges ij of the graph.

The M-polynomial of the linear phenylene G[n] is M(G[n];x,y) = 6*x^2*y^2 + 4*(n - 1)*x^2*y^3 + 4(n - 1)*x^3*y^3.

			a(1) = 24; indeed, G[1] is a hexagon; we have 6 edges, each with end vertices of degree 2; then the second Zagreb index is 6*2*2 =24.

Colin Barker, Table of n, a(n) for n = 1..1000
O. Bodroza-Pantic, I. Gutman, and S. J. Cyvin, Fibonacci numbers and algebraic structure count of some non-benzenoid conjugated polymers, The Fibonacci Quarterly, 35, 1, 1997, 75-83.
M. R. Darafsheh, Computation of topological indices of some graphs, Acta Appl. Math., 110, 2010, 1225-1235.
E. Deutsch and Sandi Klavzar, M-polynomial and degree-based topological indices, Iranian J. Math. Chemistry, 6, No. 2, 2015, 93-102.
P. Gayathri and U. Priyanka, Degree based topological indices of linear phenylene, Internat. J. of Innovative Research in Science, Engineering and Technology, 6, 8, 2017, 16986-16997.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A016873, A304157.

Subsequence of A121024.

[60*n-36 for n in 1:50] |> println # Bruno Berselli, May 09 2018

Maple
```
seq(60*n - 36, n = 1 .. 40);
```

a(n) = 60*n-36; \\ Altug Alkan, May 09 2018

Vec(12*x*(2 + 3*x)/(1 - x)^2 + O(x^40)) \\ Colin Barker, May 23 2018

a(n) = 60*n - 36.
a(n) = 12 * A016873(n-1). - Alois P. Heinz, May 09 2018
From Bruno Berselli, May 09 2018: (Start)
O.g.f.: 12*x*(2 + 3*x)/(1 - x)^2.
E.g.f.: 12*(3 - 3*exp(x) + 5*x*exp(x)).
a(n) = 2*a(n-1) - a(n-2).
a(n) = A008594(5*n-3) = A017317(6*n-4) = A072710(4*n-2) = A139245(3*n-1). (End)

A370238 a(n) = n(3n + 23)/2.

Original entry on oeis.org

0, 13, 29, 48, 70, 95, 123, 154, 188, 225, 265, 308, 354, 403, 455, 510, 568, 629, 693, 760, 830, 903, 979, 1058, 1140, 1225, 1313, 1404, 1498, 1595, 1695, 1798, 1904, 2013, 2125, 2240, 2358, 2479, 2603, 2730, 2860, 2993, 3129, 3268, 3410, 3555, 3703, 3854, 4008

Offset: 0

Torlach Rush, Feb 12 2024

a(a(1)) = A000566(a(1)). This is also true for each of the sequences provided in the formulae below; e.g., A151542(A151542(1)) = A000566(A151542(1)).

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Sela Fried, Counting r X s rectangles in nondecreasing and Smirnov words, arXiv:2406.18923 [math.CO], 2024. See p. 9.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000326, A000566, A008594, A005449, A045943, A059845, A115067, A140090, A140091, A140672, A140673, A140674, A140675, A151542, A277976.

Table[n(3n+23)/2,{n,0,48}] (* James C. McMahon, Feb 20 2024 *)

Python
```
def a(n): return n*(3*n+23)//2
```

a(n) = n*(3*n + 23)/2 = A277976(n)/2.
G.f.: x*(13-10*x)/(1-x)^3.
a(n) = A151542(n) + n.
a(n) = A140675(n) + 2*n.
a(n) = A140674(n) + 3*n.
a(n) = A140673(n) + 4*n.
a(n) = A140672(n) + 5*n.
a(n) = A059845(n) + 6*n.
a(n) = A140091(n) + 7*n.
a(n) = A140090(n) + 8*n.
a(n) = A115067(n) + 9*n.
a(n) = A045943(n) + 10*n.
a(n) = A005449(n) + 11*n.
a(n) = A000326(n) + A008594(n).
Sum_{n>=1} 1/a(n) = 823467/2769844 + sqrt(3)*Pi/69 -3*log(3)/23 = 0.2328608... - R. J. Mathar, Apr 23 2024
E.g.f.: exp(x)*x*(26 + 3*x)/2. - Stefano Spezia, Apr 26 2024

A382932 a(n) is the altitude of the Pythagorean triangle (A046083(A382931(n)), A046084(A382931(n)), A009000(A382931(n))).

Original entry on oeis.org

12, 24, 36, 48, 60, 72, 60, 84, 96, 108, 120, 132, 120, 144, 156, 120, 168, 180, 192, 204, 216, 228, 240, 180, 252, 264, 276, 240, 288, 300, 168, 312, 324, 240, 336, 348, 360, 372, 384, 396, 420, 300, 408, 360, 420, 432, 444, 456, 468, 480, 360, 492, 504, 516

Offset: 1

Felix Huber, Apr 13 2025

nonn

All terms are divisible by 12. Proof: (Start)
Let (a, b, c) be a primitive Pythagorean triple. Since gcd(a, b, c) = 1, all and only the Pythagorean triples (k*c*a, k*c*b, k*c^2) have an integer altitude h = (k*c*a*k*c*b)/(k*c^2) = k*a*b, where k is a positive integer.
With a = p^2 - q^2 and b = 2*p*q follows h = 2*k*p*q*(p^2 - q^2) = k*2*p*q*(p + q)*(p - q), where p > q > 0, gcd(p,q) = 1 and p or q is even.
It is to show that p*q*(p + q)*(p - q) is divisible by 6. Since p or q is divisible by 2, it remains to show that p*q*(p + q)*(p - q) is divisible by 3.
If 3 is a divisor of p or q, p*q is divisible by 3. If p mod 3 = 1 and q mod 3 = 2 or p mod 3 = 2 and q mod 3 = 1, then p + q is divisible by 3. If p mod 3 = q mod 3 = 1 or p mod 3 = q mod 3 = 2, then p - q is divisible by 3.
It follows that all terms are divisible by 12. (End)

			a(1) = 12 because the Pythagorean triangle (A046083(A382931(1)), A046084(A382931(1)), A009000(A382931(1))) = (A046083(7), A046084(7), A009000(7)) = (15, 20, 25) has the integer altitude 15*20/25 = 12.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A008594, A009000, A046083, A046084, A382931.

A382932:=proc(H) # All hypotenuses <= H.
    local a,b,c,k,p,q,L,M;
    L:=[];
    M:=[];
    for p from 2 to floor(sqrt(H-1)) do
        for q to min(p-1,floor(sqrt(H-p^2))) do
            if gcd(p,q)=1 and is(p-q,odd) then
                a:=p^2-q^2;
                b:=2*p*q;
                c:=p^2+q^2;
                for k to iquo(H,c) do
                    L:=[op(L),[k*c,k*max(a,b),k*a*b/c]]
                od
            fi
        od
    od;
    L:=sort(L);
    for k to nops(L) do
        if is(L[k,3],integer) then
           M:=[op(M),L[k,3]]
        fi
    od;
    return op(M)
end proc;
A382932(1075);

a(n) = A046083(A382931(n))*A046084(A382931(n))/A009000(A382931(n)).

A383488 Numbers k that have at least one divisor d_i(k) for which a divisor d_j(k) exists such that d_i(k) < d_j(k) < sigma(d_i(k)).

Original entry on oeis.org

12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 130, 132, 138, 140, 144, 150, 154, 156, 160, 162, 168, 170, 174, 176, 180, 186, 189, 192, 196, 198, 200, 204, 208, 210, 216

Offset: 1

Felix Huber, May 03 2025

nonn

Numbers k (without multiplicity) that are multiples of lcm(c,i), where c is any composite and i is any integer from [c + 1, sigma(c) - 1].

			All multiples of 12 (A008594) are terms because 12 has the divisors 4 and 6 where sigma(4) = 7 > 6.
All multiples of 18 (A008600) are terms because 18 has the divisors 6 and 9 where sigma(6) = 12 > 9.
All multiples of 20 (A008602) are terms because 20 has the divisors 4 and 5 where sigma(4) = 7 > 5.

Felix Huber, Table of n, a(n) for n = 1..10000

Supersequence of A008594, A008600, A008602, A008606, A044102, A135628.

Cf. A000203, A002808, A027750, A109042, A383360, A383489.

with(NumberTheory):
A383488:=proc(n)
    option remember;
    local k,i,L;
    if n=1 then
        12
    else
        for k from procname(n-1)+1 do
            L:=Divisors(k);
            for i to nops(L)-1 do
                if sigma(L[i])>L[i+1] then
                    return k
                fi
            od
        od
    fi;
end proc;
seq(A383488(n),n=1..57);

A062902 Number and its reversal are both multiples of 12.

Original entry on oeis.org

0, 48, 84, 216, 252, 276, 408, 420, 444, 468, 480, 612, 636, 672, 696, 804, 828, 840, 864, 888, 2100, 2112, 2124, 2136, 2148, 2160, 2172, 2184, 2196, 2304, 2316, 2328, 2340, 2352, 2364, 2376, 2388, 2508, 2520, 2532, 2544, 2556, 2568, 2580, 2592, 2700, 2712

Offset: 1

Amarnath Murthy, Jul 01 2001

Numbers divisible by 12 with reversal divisible by 4. - Robert Israel, May 04 2025

			216 and 612 are both multiples of 12.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A062897 (2), A062898 (4), A062899 (6), A062900 (8), A062901 (7), A062903 (13), A062904 (14), A062905 (15), A062906 (17), A062907 (19).

Cf. A004086, A008594.

n := 12; stop := 2800; m := 0; while m < stop do rev := int_reverse(m); if rev mod n = 0 then write(m," "); end; inc(m,n); end;

rev:= proc(n) local L,i;
  L:= convert(n,base,10);
  add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
select(t -> rev(t) mod 12 = 0, [seq(i,i=0..3000,12)]); # Robert Israel, May 04 2025

Corrected and extended by Dean Hickerson, Jul 06 2001

Offset changed by Georg Fischer, Sep 08 2022

A063669 Hypotenuses of reciprocal Pythagorean triangles: number of solutions to 1/(12n)^2 = 1/b^2 + 1/c^2 [with b >= c > 0]; also number of values of A020885 (with repetitions) which divide n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5

Offset: 1

Henry Bottomley, Jul 28 2001

nonn

Primitive reciprocal Pythagorean triangles 1/a^2 = 1/b^2 + 1/c^2 have a=fg, b=ef, c=eg where e^2 = f^2 + g^2; i.e., e,f,g represent the sides of primitive Pythagorean triangles. But the product of the two legs of primitive Pythagorean triangles are multiples of 12 and so the reciprocal of hypotenuses of reciprocal Pythagorean triangles are always multiples of 12 (A008594).

			a(1)=1 since 1/(12*1)^2 = 1/12^2 = 1/15^2 + 1/20^2;
a(70)=6 since 1/(12*70)^2 = 1/840^2 = 1/875^2 + 1/3000^2 = 1/888^2 + 1/2590^2 = 1/910^2 + 1/2184^2 = 1/952^2 + 1/1785^2 = 1/1050^2 + 1/1400^2 = 1/1160^2 + 1/1218^2.
Looking at A020885, 1 is divisible by 1, while 70 is divisible by 1, 5, 10, 14, 35 and again 35.

Cf. A046080, A063664, A063665, A063014.

A309118 Number of tiles added at iteration n when successively, layer by layer, building a symmetric patch of a rhombille tiling around a central star of six rhombs.

Original entry on oeis.org

6, 6, 12, 18, 24, 24, 36, 30, 48, 36, 60, 42, 72, 48, 84, 54, 96, 60, 108, 66, 120, 72, 132, 78, 144, 84, 156, 90, 168, 96, 180, 102, 192, 108, 204, 114, 216, 120, 228, 126, 240, 132, 252, 138, 264, 144, 276, 150, 288, 156, 300, 162, 312, 168, 324, 174, 336

Offset: 1

Felix Fröhlich, Jul 13 2019

			See illustration in Fröhlich, 2019.

Colin Barker, Table of n, a(n) for n = 1..1000
Felix Fröhlich, Layers in the rhombille tiling, 2019.
Wikipedia, Rhombille tiling
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008588, A008594.

Cf. A242128 (5-fold, Star), A242129 (5-fold, Sun), A242888 (7-fold, Star), A242889 (7-fold, Sun), A242890 (8-fold, Star), A242891 (8-fold, Sun), A242892 (9-fold, Star), A242893 (9-fold, Sun), A242894 (Kite and dart, Star), A242895 (Kite and dart, Sun).

I:=[6,6,12,18,24,24]; [n le 6 select I[n] else 2*Self(n-2)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, Jul 16 2019

Join[{6, 6}, LinearRecurrence[{0, 2, 0, -1}, {12, 18, 24, 24}, 60]] (* Vincenzo Librandi, Jul 16 2019 *)

a(n) = if(n<3, 6, if(n%2==0, 6*((n+2)/2), 12*((n-1)/2)))

Vec(6*x*(1 + x + x^3 + x^4 - x^5) / ((1 - x)^2*(1 + x)^2) + O(x^40)) \\ Colin Barker, Jul 13 2019

a(2*n+1) = A008594(n).
a(2*n) = A008588(n+1) for n > 1.
From Colin Barker, Jul 13 2019: (Start)
G.f.: 6*x*(1 + x + x^3 + x^4 - x^5) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>6.
(End)

A332019 The number of cells added in the n-th generation of the following procedure: start by coloring any triangle on the snub square tiling, then repeatedly color every cell that shares a vertex with a colored cell.

Original entry on oeis.org

1, 9, 21, 35, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228, 240, 252, 264, 276, 288, 300, 312, 324, 336, 348, 360, 372, 384, 396, 408, 420, 432, 444, 456, 468, 480, 492, 504, 516, 528, 540, 552, 564, 576, 588, 600, 612, 624, 636

Offset: 1

Peter Kagey, Feb 04 2020

Peter Kagey, Table of n, a(n) for n = 1..1000
Code Golf Stack Exchange, Concentric rings on a snub square tiling.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A008594.

A296368 is the analogous sequence when instead coloring every cell that shares a side with a colored cell.

a(n) = 12*(n - 1) for n > 4.
From Stefano Spezia, Feb 05 2020: (Start)
G.f.: x*(1 + 7*x + 4*x^2 + 2*x^3  - x^4 - x^5)/(-1 + x)^2.
a(n) = 2*a(n-1) - a(n-2) for n > 6.
(End)

A333814 Multiples of 12 whose sum of digits is 12.

Original entry on oeis.org

48, 84, 156, 192, 228, 264, 336, 372, 408, 444, 480, 516, 552, 624, 660, 732, 804, 840, 912, 1056, 1092, 1128, 1164, 1236, 1272, 1308, 1344, 1380, 1416, 1452, 1524, 1560, 1632, 1704, 1740, 1812, 1920, 2028, 2064, 2136, 2172, 2208, 2244, 2280, 2316, 2352, 2424

Offset: 1

Bernard Schott, Apr 06 2020

If m is a term, 10*m is also a term.

			732 = 12 * 61 and 7 + 3 + 2 = 12, hence 732 is a term.

Intersection of A235151 (sum of digits = 12) and A008594 (multiples of 12).
Multiples of k whose sum of digits = k: A011557 (k=1), A069537 (k=2), A052217 (k=3), A063997 (k=4), A069540 (k=5), A062768 (k=6), A063416 (k=7), A069543 (k=8), A052223 (k=9), A333834 (k=10), A283742 (k=11), this sequence (k=12), A283737 (k=13).
Cf. A008594 (multiples of 12), A235151 (sum of digits = 12).
Cf. A057147 (a(n) = n times sum of digits of n).

Select[12 * Range[200], Plus @@ IntegerDigits[#] == 12 &] (* Amiram Eldar, Apr 06 2020 *)

is(n)=sumdigits(n)==12 && n%4==0 \\ Charles R Greathouse IV, Apr 07 2020

a(n) ~ A235151(n). - Charles R Greathouse IV, Apr 07 2020

A144598 Christoffel word of slope 5/7.

Original entry on oeis.org

0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0

Offset: 0

N. J. A. Sloane, Jan 13 2009

The path is on the slope after 0, 12, 24, 36, 48,... (A008594) steps, which gives the C-finite recurrence. - R. J. Mathar, May 28 2025

See A144595 for further details.

a(n) = a(n-12). - R. J. Mathar, May 28 2025

G.f.: -x^2*(1+x^2+x^5+x^7+x^9) / ( (x-1)*(1+x+x^2)*(1+x)*(1-x+x^2)*(1+x^2)*(x^4-x^2+1) ). - R. J. Mathar, May 28 2025

cp's OEIS Frontend

A304158 a(n) is the second Zagreb index of the linear phenylene G[n], defined pictorially in the Darafsheh reference (Fig. 3).

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A370238 a(n) = n*(3*n + 23)/2.

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A382932 a(n) is the altitude of the Pythagorean triangle (A046083(A382931(n)), A046084(A382931(n)), A009000(A382931(n))).

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A383488 Numbers k that have at least one divisor d_i(k) for which a divisor d_j(k) exists such that d_i(k) < d_j(k) < sigma(d_i(k)).

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A062902 Number and its reversal are both multiples of 12.

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ARIBAS

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A063669 Hypotenuses of reciprocal Pythagorean triangles: number of solutions to 1/(12n)^2 = 1/b^2 + 1/c^2 [with b >= c > 0]; also number of values of A020885 (with repetitions) which divide n.

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A309118 Number of tiles added at iteration n when successively, layer by layer, building a symmetric patch of a rhombille tiling around a central star of six rhombs.

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Magma

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A332019 The number of cells added in the n-th generation of the following procedure: start by coloring any triangle on the snub square tiling, then repeatedly color every cell that shares a vertex with a colored cell.

A370238 a(n) = n(3n + 23)/2.