A093518 -id:A093518 - cp's OEIS frontend

A052343 Number of ways to write n as the unordered sum of two triangular numbers (zero allowed).

1, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 0, 1, 0, 1, 2, 1, 0, 1, 1, 0, 1, 1, 1, 1, 2, 0, 0, 1, 0, 2, 1, 1, 1, 0, 0, 2, 1, 0, 1, 2, 0, 1, 1, 0, 2, 0, 0, 0, 2, 2, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 0, 1, 1, 0, 3, 0, 1, 1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 2, 0, 0, 1, 0, 1, 1, 1

Offset: 0

Christian G. Bower, Jan 23 2000

Number of ways of writing n as a sum of a square and twice a triangular number (zeros allowed). - Michael Somos, Aug 18 2003
a(A020757(n))=0; a(A020756(n))>0; a(A119345(n))=1; a(A118139(n))>1. - Reinhard Zumkeller, May 15 2006
Also, number of ways to write 4n+1 as the unordered sum of two squares of nonnegative integers. - Vladimir Shevelev, Jan 21 2009
The average value of a(n) for n <= x is Pi/4 + O(1/sqrt(x)). - Vladimir Shevelev, Feb 06 2009

			G.f. = 1 + x + x^2 + x^3 + x^4 + 2*x^6 + x^7 + x^9 + x^10 + x^11 + ...

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Vladimir Shevelev, Binary additive problems: recursions for numbers of representations, arXiv:0901.3102 [math.NT], 2009-2013.
Vladimir Shevelev, Binary additive problems: theorems of Landau and Hardy-Littlewood type, arXiv:0902.1046 [math.NT], 2009-2012.

Cf. A000217, A052344, A052345 (greedy inverse), A052346, A052347, A052348, A053587, A056303, A056304.
Cf. A053692, A093518, A121444, A259285, A259287, A260415, A260516.
Cf. A005369, A010052.

a052343 = (flip div 2) . (+ 1) . a008441
-- Reinhard Zumkeller, Jul 25 2014

A052343 := proc(n)
    local a,t1idx,t2idx,t1,t2;
    a := 0 ;
    for t1idx from 0 do
        t1 := A000217(t1idx) ;
        if t1 > n then
            break;
        end if;
        for t2idx from t1idx do
            t2 := A000217(t2idx) ;
            if t1+t2 > n then
                break;
            elif t1+t2 = n then
                a := a+1 ;
            end if;
        end do:
    end do:
    a ;
end proc: # R. J. Mathar, Apr 28 2020

Length[PowersRepresentations[4 # + 1, 2, 2]] & /@ Range[0, 101] (* Ant King, Dec 01 2010 *)
d1[k_]:=Length[Select[Divisors[k],Mod[#,4]==1&]];d3[k_]:=Length[Select[Divisors[k],Mod[#,4]==3&]];f[k_]:=d1[k]-d3[k];g[k_]:=If[IntegerQ[Sqrt[4k+1]],1/2 (f[4k+1]+1),1/2 f[4k+1]];g[#]&/@Range[0,101] (* Ant King, Dec 01 2010 *)
a[ n_] := Length @ Select[ Table[ Sqrt[n - i - i^2], {i, 0, Quotient[ Sqrt[4 n + 1] - 1, 2]}], IntegerQ]; (* Michael Somos, Jul 28 2015 *)
a[ n_] := Length @ FindInstance[ {j >= 0, k >= 0, j^2 + k^2 + k == n}, {k, j}, Integers, 10^9]; (* Michael Somos, Jul 28 2015 *)

{a(n) = if( n<0, 0, sum(i=0, (sqrtint(4*n + 1) - 1)\2, issquare(n - i - i^2)))}; /* Michael Somos, Aug 18 2003 */

a(n) = ceiling(A008441(n)/2). - Reinhard Zumkeller, Nov 03 2009
G.f.: (Sum_{k>=0} x^(k^2 + k)) * (Sum_{k>=0} x^(k^2)). - Michael Somos, Aug 18 2003
Recurrence: a(n) = Sum_{k=1..r(n)} r(2n-k^2+k) - C(r(n),2) - a(n-1) - a(n-2) - ... - a(0), n>=1,a (0)=1, where r(n)=A000194(n+1) is the nearest integer to square root of n+1. For example, since r(6)=3, a(6) = r(12) + r(10) + r(6) - C(3,2) - a(5) - ... - a(0) = 4 + 3 + 3 - 3 - 0 - 1 - 1 - 1 - 1 - 1 = 2. - Vladimir Shevelev, Feb 06 2009
a(n) = A025426(8n+2). - Max Alekseyev, Mar 09 2009
a(n) = (A002654(4n+1) + A010052(4n+1)) / 2. - Ant King, Dec 01 2010
a(2*n + 1) = A053692(n). a(4*n + 1) = A259287(n). a(4*n + 3) = A259285(n). a(6*n + 1) = A260415(n). a(6*n + 4) = A260516(n). - Michael Somos, Jul 28 2015
a(3*n) = A093518(n). a(3*n + 1) = A121444(n). a(9*n + 2) = a(n). a(9*n + 5) = a(9*n + 8) = 0. - Michael Somos, Jul 28 2015
Convolution of A005369 and A010052. - Michael Somos, Jul 28 2015

A093519 Numbers with no representation as the sum of two (not necessarily distinct) generalized pentagonal numbers.

Original entry on oeis.org

11, 18, 21, 25, 32, 39, 43, 46, 49, 54, 60, 65, 67, 68, 74, 76, 81, 87, 88, 90, 95, 98, 106, 109, 111, 113, 116, 120, 123, 125, 130, 136, 137, 142, 144, 153, 158, 159, 163, 164, 165, 172, 173, 175, 179, 182, 186, 193, 197, 201, 204, 205, 207, 208, 214, 219, 220

Offset: 1

Jon Perry, Mar 29 2004

nonn

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

Cf. A001318 (generalized pentagonal numbers), A093518.

Cf. A204382.

GP:= [0, seq(op([m*(3*m-1)/2, m*(3*m+1)/2]), m=1..50)]:
N:= GP[-1]:
V:= Array(0..N, datatype=integer[4]):
for i from 1 to nops(GP) do
for j from 1 to i do
   r:= GP[i]+GP[j];
   if r > N then break fi;
   V[r]:= V[r]+1
od od:
select(t -> V[t] = 0, [$0..N]); # Robert Israel, Feb 26 2025

Definition clarified by Robert Israel, Feb 26 2025

A290943 Number of ways to write n as an ordered sum of 3 generalized pentagonal numbers (A001318).

Original entry on oeis.org

1, 3, 6, 7, 6, 6, 7, 12, 12, 12, 9, 6, 12, 12, 18, 13, 12, 18, 12, 18, 12, 13, 18, 12, 24, 12, 12, 24, 21, 30, 12, 18, 18, 12, 24, 18, 19, 18, 24, 24, 18, 24, 36, 24, 18, 19, 18, 24, 24, 30, 18, 12, 36, 30, 24, 21, 18, 36, 24, 36, 24, 12, 36, 36, 36, 18, 25, 30, 24, 24, 24, 30, 24, 36, 30, 24

Offset: 0

Ilya Gutkovskiy, Aug 14 2017

Conjecture: every number is the sum of at most k - 4 generalized k-gonal numbers (for k >= 8).

In 1830, Legendre showed that for each integer m>4 every integer N >= 28*(m-2)^3 can be written as the sum of five m-gonal numbers. In 1994 R. K. Guy proved that each natural number is the sum of three generalized pentagonal numbers. In a 2016 paper Zhi-Wei Sun proved that each natural number is the sum of four octagonal numbers. - Zhi-Wei Sun, Oct 03 2020

			a(6) = 7 because we have [5, 1, 0], [5, 0, 1], [2, 2, 2], [1, 5, 0], [1, 0, 5], [0, 5, 1] and [0, 1, 5].

Robert Israel, Table of n, a(n) for n = 0..10000
Ilya Gutkovskiy, Extended graphical example
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
Eric Weisstein's World of Mathematics, Pentagonal Number
Index to sequences related to polygonal numbers

Cf. A001318, A002175, A008443, A080995, A093518, A093519, A255350, A255934, A256132, A256171, A280718.

N:= 100;
bds:= [fsolve(k*(3*k-1)/2 = N)];
G:= add(x^(k*(3*k-1)/2),k=floor(min(bds))..ceil(max(bds)))^3:
seq(coeff(G,x,n),n=0..N); # Robert Israel, Aug 16 2017

nmax = 75; CoefficientList[Series[Sum[x^(k (3 k - 1)/2), {k, -nmax, nmax}]^3, {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[Sum[x^((6 k^2 + 6 k + (-1)^(k + 1) (2 k + 1) + 1)/16), {k, 0, nmax}]^3, {x, 0, nmax}], x]
nmax = 75; CoefficientList[Series[EllipticTheta[4, 0, x^3]^3/QPochhammer[x, x^2]^3, {x, 0, nmax}], x]

G.f.: (Sum_{k=-infinity..infinity} x^(k*(3*k-1)/2))^3.
G.f.: (Sum_{k>=0} x^A001318(k))^3.
G.f.: Product_{n >= 1} ( (1 - q^(3*n))/(1 - q^n + q^(2*n)) )^3. - Peter Bala, Jan 04 2025

A381535 a(n) is the least nonnegative number that can be represented as the sum of two (not necessarily distinct) generalized pentagonal numbers in exactly n ways.

Original entry on oeis.org

11, 0, 2, 27, 92, 352, 1002, 16927, 2302, 7827, 25052, 220052, 13352, 1487552, 101752, 195677, 85177, 137532552, 173577

Offset: 0

Robert Israel, Feb 26 2025

a(n) is the least k >= 0 such that A093518(k) = n.
a(17) > 5.4 * 10^7 if it exists.
From Pontus von Brömssen, Feb 28 2025: (Start)
a(19) > 3*10^9 if it exists.
After a(19), the following are all terms below 3*10^9:
   n |    a(n)
  ---+-----------
|     333802
|    4891927
|  391438802
|    2543802
|     494027
|   55039427
|    3764827
|    8345052
|    4339427
|    2737177
| 1375985677
|    6422352
|  429902552
|   12350677
|   85573502
|  108485677
|   94120677
|   29014077
|  733363177
|  120983227
|  308766927
|  160558802
| 2353016927
|  101275552
| 2139337552
|  344336877
|  725351927
| 1073520852
(End)

			a(3) = 27 because 27 = 1 + 26 = 5 + 22 = 12 + 15 has 3 representations as the sum of two generalized pentagonal numbers, and no smaller number works.

Cf. A001318, A093518.

GP:= [0,seq(op([m*(3*m-1)/2, m*(3*m+1)/2]),m=1..2000)]:
N:= GP[-1]:
V:= Array(0..N, datatype=integer[4]):
for i from 1 to nops(GP) do
for j from 1 to i do
   r:= GP[i]+GP[j];
   if r > N then break fi;
   V[r]:= V[r]+1
od od:
W:= Array(0..16): count:= 0:
for i from 1 to N while count < 17 do
  v:= V[i]; if v <= 16 and W[v] = 0 then W[v]:= i; count:= count + 1 fi
od:
W[1]:= 0:
convert(W,list);

A093518(a(n)) = n.

a(17)-a(18) from Pontus von Brömssen, Feb 28 2025

cp's OEIS Frontend

A052343 Number of ways to write n as the unordered sum of two triangular numbers (zero allowed).

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A093519 Numbers with no representation as the sum of two (not necessarily distinct) generalized pentagonal numbers.

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A290943 Number of ways to write n as an ordered sum of 3 generalized pentagonal numbers (A001318).

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A381535 a(n) is the least nonnegative number that can be represented as the sum of two (not necessarily distinct) generalized pentagonal numbers in exactly n ways.

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