A133263 -id:A133263 - cp's OEIS frontend

A152948 a(n) = (n^2 - 3*n + 6)/2.

2, 2, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227, 1277, 1328, 1380

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 15 2008

a(1) = 2; then add 0 to the first number, then 1, 2, 3, 4, ... and so on.
Essentially the same as A022856, A089071 and A133263. - R. J. Mathar, Dec 19 2008
First differences are A001477.
From Vladimir Shevelev, Jan 20 2014: (Start)
If we ignore the zero polygonal numbers, then for n >= 3, a(n) is the minimal k such that the k-th n-gonal number is a sum of two n-gonal numbers (see formula and example).
If the zero polygonal numbers are ignored, then for n >= 4, the a(n)-th n-gonal number is a sum of the (a(n)-1)-th n-gonal number and the (n-1)-th n-gonal number. (End)
Numbers m such that 8m - 15 is a square. - Bruce J. Nicholson, Jul 24 2017

			a(7)=17. This means that the 17th (positive) heptagonal number 697 (cf. A000566) is the smallest heptagonal number which is a sum of two (positive) heptagonal numbers. We have 697 = 616 + 81 with indices 17, 16, 6 in A000566. - _Vladimir Shevelev_, Jan 20 2014

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Marilena Barnabei, Flavio Bonetti, Niccolò Castronuovo, and Matteo Silimbani, Permutations avoiding a simsun pattern, The Electronic Journal of Combinatorics (2020) Vol. 27, Issue 3, P3.45.
E. R. Berlekamp, A contribution to mathematical psychometrics, Unpublished Bell Labs Memorandum, Feb 08 1968 [Annotated scanned copy]
Kyu-Hwan Lee and Se-jin Oh, Catalan triangle numbers and binomial coefficients, arXiv:1601.06685 [math.CO], 2016.
Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000124, A000217, A000566, A001477, A022856, A089071, A133263, A152947.

Magma
```
[ (n^2-3*n+6)/2: n in [1..60] ];
```

Array[(#^2 - 3 # + 6)/2 &, 54] (* or *) Rest@ CoefficientList[Series[-x (2 - 4 x + 3 x^2)/(x - 1)^3, {x, 0, 54}],x] (* Michael De Vlieger, Mar 25 2020 *)

a(n)=(n^2-3*n+6)/2 \\ Charles R Greathouse IV, Sep 28 2015

[2+binomial(n,2) for n in range(0, 54)] # Zerinvary Lajos, Mar 12 2009

a(n) = a(n-1) + n-2 (with a(1)=2). - Vincenzo Librandi, Nov 26 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -x*(2 - 4*x + 3*x^2) /  (x-1)^3. - R. J. Mathar, Oct 30 2011
Sum_{n>=1} 1/a(n) = 1/2 + 2*Pi*tanh(sqrt(15)*Pi/2)/sqrt(15). - Amiram Eldar, Dec 13 2022
E.g.f.: exp(x)*(6 - 2*x + x^2)/2 - 3. - Stefano Spezia, Nov 14 2024

A180681 T(n,k) is the sum of the hook lengths over the partitions of n with exactly k parts.

Original entry on oeis.org

1, 3, 3, 6, 5, 6, 10, 16, 8, 10, 15, 23, 22, 12, 15, 21, 47, 44, 30, 17, 21, 28, 62, 74, 56, 40, 23, 28, 36, 104, 115, 114, 71, 52, 30, 36, 45, 130, 196, 162, 139, 89, 66, 38, 45, 55, 195, 268, 286, 227, 169, 110, 82, 47, 55, 66, 235, 395, 407, 369, 269, 204, 134, 100, 57, 66

Offset: 1

Wouter Meeussen, Sep 16 2010

Row sums equal A066183 ('Total sum of squares of parts in all partitions of n').
From Omar E. Pol, Mar 20 2018: (Start)
Both column 1 and leading diagonal give A000217, n >= 1.
Both A206561 and A299768 have the same row sums as this triangle.
Apparently the second diagonal gives A133263 without the first term. (End)

			T(5,3) = 22 since the partitions of 5 in 3 parts are 221 and 311, with hook lengths {{2,4}, {1,3}, {1}} and {{1,2,5}, {2}, {1}} summing to 22.
Triangle T(n,k) begins:
   1;
   3,   3;
   6,   5,   6;
  10,  16,   8,  10;
  15,  23,  22,  12,  15;
  21,  47,  44,  30,  17,  21;
  28,  62,  74,  56,  40,  23,  28;
  36, 104, 115, 114,  71,  52,  30, 36;
  45, 130, 196, 162, 139,  89,  66, 38, 45;
  55, 195, 268, 286, 227, 169, 110, 82, 47, 55;

Alois P. Heinz, Rows n = 1..200, flattened

Cf. A000217, A008284, A066183, A133263, A206561, A299768.

T(2n-1,n) gives A301499.

f:= n-> (n-1)*n/2:
b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n+f(n)],
      b(n, i-1)+(p-> p+[0, p[1]*(n+f(i))])(b(n-i, min(n-i, i))))
    end:
T:= (n, k)-> (p-> p[1]*(n+f(k))+p[2])(b(n-k, min(n-k, k))):
seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Mar 20 2018

(*Needs["DiscreteMath`Combinatorica`"]; hooklength[(p_)?PartitionQ] := Block[{ferr = (PadLeft[1 + 0*Range[ #1], Max[p]] & ) /@ p}, DeleteCases[(Rest[FoldList[Plus, 0, #1]] & ) /@ ferr + Reverse /@ Reverse[Transpose[(Rest[FoldList[Plus, 0, #1]] & ) /@ Reverse[Reverse /@ Transpose[ferr]]]], 0, {2}] - 1]; partitionexact[n_, m_] := TransposePartition /@ (Prepend[ #1, m] & ) /@ Partitions[n - m, m] *); Table[Tr[ Tr[ Flatten[hooklength[ # ]]] &/@ partitionexact[n,k] ] ,{n,16},{k,n}]
(* Second program: *)
Table[p = IntegerPartitions[n, {k}]; Total@Table[y = Table[Boole[p[[l]][[i]] >= j], {i, k}, {j, n}]; Total[Table[Total[{y[[i, j ;; n]], y[[i + 1 ;; k, j]]}, 2], {i, k}, {j, n}], 2], {l, Length[p]}], {n, 11}, {k, n}] // Flatten (* Robert Price, Jun 19 2020 *)
f[n_] := n(n-1)/2;
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, n + f[n]}, b[n, i - 1] + Function[p, p + {0, p[[1]] (n + f[i])}][b[n - i, Min[n - i, i]]]];
T[n_, k_] := Function[p, p[[1]] (n + f[k]) + p[[2]]][b[n-k, Min[n-k, k]]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Dec 12 2020, after Alois P. Heinz *)

A228446 a(n) = smallest prime p such that 2n+1 = p + x(x+1) for some positive integer x, or -1 if no such prime exists.

Original entry on oeis.org

3, 5, 3, 5, 7, 3, 5, 7, 19, 3, 5, 7, 17, 11, 3, 5, 7, 19, 11, 13, 3, 5, 7, 31, 11, 13, 37, 3, 5, 7, 23, 11, 13, 29, 17, 3, 5, 7, 61, 11, 13, 31, 17, 19, 3, 5, 7, 43, 11, 13, 103, 17, 19, 109, 3, 5, 7, 29, 11, 13, 53, 17, 19, 41, 23, 3, 5, 7, 31, 11, 13, 37

Offset: 2

Bill McEachen, Oct 26 2013

Based on Sun's conjecture 1.4 in the paper referenced below.
The plot shows an ever-widening band of sawtooth shape.  New maxima values will include sequence members larger than the largest prime factor of the original n.  For example when n = 21 with prime factors 3 and 7, and a(10) = 19.
a(A000124(n)) = 3; a(A133263(n)) = 5; a(A167614(n)) = 7. - Reinhard Zumkeller, Mar 12 2014

			21 = 19+1*2 where no solution exists using p = 2, 3, 5, 7, 11, 13, 17. So a(10) = 19.
51 = 31+4*5 where no lower odd prime provides a solution. So a(25) = 31.

Z. W. Sun, On sums of primes and triangular numbers, Journal of Combinatorics and Number Theory 1(2009), no. 1, 65-76. (See Conjecture 1.4.)

T. D. Noe, Table of n, a(n) for n = 2..1000
Z. W. Sun, On sums of primes and triangular numbers, arXiv:0803.3737 [math.NT], 2008-2009.
Christian Krause, et al, A mined LODA assembly source for this sequence (conjectured)

Cf. A010051, A002378, A000217.

a228446 n = head
   [q | let m = 2 * n + 1,
        q <- map (m -) $ reverse $ takeWhile (< m) $ tail a002378_list,
        a010051 q == 1]
-- Reinhard Zumkeller, Mar 12 2014

nn = 14; ob = Table[n*(n+1), {n, nn}]; Table[p = Min[Select[n - ob, # > 0 && PrimeQ[#] &]]; p, {n, 5, ob[[-1]], 2}] (* T. D. Noe, Oct 27 2013 *)

a(n) = {oddn = 2*n+1; x = oddn; while (! isprime(oddn - x*(x+1)), x--); oddn - x*(x+1);} \\ Michel Marcus, Oct 27 2013

Entry revised by N. J. A. Sloane, Nov 11 2020 (including addition of escape clause).

A367849 Lexicographically earliest infinite sequence of positive integers such that for each n, the values in a path of locations starting from any i=n are all distinct, where jumps are allowed from location i to i+a(i).

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 1, 4, 1, 1, 2, 5, 3, 1, 1, 4, 6, 1, 1, 5, 1, 1, 7, 1, 6, 1, 1, 2, 1, 8, 7, 1, 1, 2, 1, 3, 1, 9, 4, 1, 1, 2, 5, 3, 1, 1, 10, 6, 1, 1, 2, 1, 3, 7, 1, 4, 11, 1, 1, 5, 8, 1, 1, 2, 6, 3, 1, 12, 9, 1, 7, 1, 1, 2, 1, 3, 1, 10, 4, 13, 1, 1, 5, 1, 1, 2, 1, 11, 1, 1, 2, 1, 14, 1, 1, 2, 1, 3

Offset: 1

Neal Gersh Tolunsky, Dec 08 2023

nonn

Consider each index i as a location from which one can jump a(i) terms forward. No starting index can reach the same value more than once by forward jumps.
The value a(i) at the starting index is not part of the path (and thus allows a(2)=1).
The indices of first occurrences are given by A133263 (essentially triangular numbers + 2).
Changing the definition so that jumps are allowed only from location i to i-a(i) gives A002260.

			We can see, for example, that the terms reachable by jumping forward continuously from i=1 are all distinct (and in this case are just the positive integers):
  1, 1, 2, 1, 3, 1, 1, 4, 1, 1, 2, 5
  *->1->2---->3------->4---------->5
Beginning at i=9 and jumping forward continuously, we get the sequence 1,2,3,4,5,6,7,9 (in which all terms are likewise distinct).

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000
Thomas Scheuerle, Plot of a(1..100000)^2. It is conjectured that only the topmost straight line in this plot will extend into infinity.
Thomas Scheuerle, Partial view of a(1..80000)^2 equalized to Y axis by shear mapping. This shows the different lengths of the increasing sequences.
Neal Gersh Tolunsky, Ordinal transform of 20000 terms
Neal Gersh Tolunsky, First differences of 20000 terms

Cf. A367467, A367832, A133263 (index of first occurrences), A362248.

function a = A367849( max_n )
    a = zeros(1,max_n); j = find(a == 0,1);
    while ~isempty(j)
        a(j) = 1; k = 1;
        if j+k < max_n
            while a(j+k) == 0
                a(j+k) = k;
                if j+2*k-1 < max_n
                    j = j+(k-1); k = k+1;
                else
                    break;
                end
            end
        end
        j = find(a == 0,1);
    end
end % Thomas Scheuerle, Dec 09 2023

a(A133263(n)) = n + 1.

A238531 Expansion of (1 - x + x^2)^2 / (1 - x)^3 in powers of x.

Original entry on oeis.org

1, 1, 3, 5, 8, 12, 17, 23, 30, 38, 47, 57, 68, 80, 93, 107, 122, 138, 155, 173, 192, 212, 233, 255, 278, 302, 327, 353, 380, 408, 437, 467, 498, 530, 563, 597, 632, 668, 705, 743, 782, 822, 863, 905, 948, 992, 1037, 1083, 1130, 1178, 1227, 1277, 1328, 1380

Offset: 0

Michael Somos, Feb 28 2014

Essentially the same as A152948, A133263 and A089071. - R. J. Mathar, Mar 30 2014

			G.f. = 1 + x + 3*x^2 + 5*x^3 + 8*x^4 + 12*x^5 + 17*x^6 + 23*x^7 + 30*x^8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A133263.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x+x^2)^2/(1-x)^3)); // G. C. Greubel, Aug 07 2018

a[ n_] := (n^2 - n) / 2 + If[ n == 0 || n == 1, 1, 2];
CoefficientList[Series[(1-x+x^2)^2/(1-x)^3, {x, 0, 50}], x] (* G. C. Greubel, Aug 07 2018 *)

{a(n) = (n^2 - n) / 2 + 2 - (n==0) - (n==1)};

{a(n) = if( n<0, n = 1-n); polcoeff( (1 - x + x^2)^2 / (1 - x)^3 + x * O(x^n), n)};

Euler transform of length 6 sequence [1, 2, 2, 0, 0, -2].
Binomial transform of [1, 0, 2, -2, 3, -4, 5, -6, ...].
a(n) = p(-1) where p(x) is the unique degree-n polynomial such that p(k) = a(k) for k = 0, 1, ..., n.
G.f.: (1 - x + x^2)^2 / (1 - x)^3.
a(n) = a(1 - n) for all n in Z.
a(n + 1) = A133263(n) if n>=0. a(n) = (n^2 - n) / 2 + 2 unless n=0 or n=1.
(1 + x^2 + x^3 + x^4 + ...)*(1 + x + 2x^2 + 3x^3 + 4x^4 + ...) = (1 + x + 3x^2 + 5x^3 + 8x^4 + 12x^5 + ...). - Gary W. Adamson, Jul 27 2010

cp's OEIS Frontend

A152948 a(n) = (n^2 - 3*n + 6)/2.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A180681 T(n,k) is the sum of the hook lengths over the partitions of n with exactly k parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A228446 a(n) = smallest prime p such that 2*n+1 = p + x*(x+1) for some positive integer x, or -1 if no such prime exists.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A367849 Lexicographically earliest infinite sequence of positive integers such that for each n, the values in a path of locations starting from any i=n are all distinct, where jumps are allowed from location i to i+a(i).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

MATLAB

Formula

A238531 Expansion of (1 - x + x^2)^2 / (1 - x)^3 in powers of x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A228446 a(n) = smallest prime p such that 2n+1 = p + x(x+1) for some positive integer x, or -1 if no such prime exists.